Every calendar in the world solves the same arithmetic problem, and there are only three solutions. A lunation is about 29.53 days and a tropical year about 365.2422 days, and the second is not a whole multiple of the first. A calendar can therefore follow the Moon and let its year drift against the sun, follow the sun and let its months lose contact with the Moon, or follow both and insert an extra month at intervals to keep them together.
The Jewish calendar takes the third path. Twelve lunar months of 29 or 30 days give 354, eleven short of the solar year, and a thirteenth month is added in seven years out of every nineteen on a fixed cycle, which keeps Passover in the spring as the Torah requires. The nineteen-year relation is the same one Greek astronomy knew as the Metonic cycle, since 235 lunations come to within a couple of hours of nineteen solar years.
What distinguishes it from other lunisolar calendars is that it is calculated rather than observed. The present fixed calendar, with its rules of postponement that prevent certain holy days from falling on certain weekdays, replaced a system of monthly observation and proclamation by the court, and the transition is generally placed in the first millennium of the common era though its stages are debated. The result is a calendar that can be computed centuries ahead, which is unusual among lunisolar systems.
The era counts from a calculated date of creation, so the year number is far larger than in the civil calendar, and the year begins in the autumn at Rosh Hashanah while the months are numbered from the spring, which is a genuine double convention rather than an inconsistency.